The polymath blog

July 12, 2012

Minipolymath4 project: IMO 2012 Q3

Filed under: research — Terence Tao @ 10:00 pm
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This post marks the official opening of the mini-polymath4 project to solve a problem from the 2012 IMO.  This time, I have selected Q3, which has an interesting game-theoretic flavour to it.

Problem 3.   The liar’s guessing game is a game played between two players A and B.  The rules of the game depend on two positive integers k and n which are known to both players.

At the start of the game, A chooses two integers x and N with 1 \leq x \leq N.  Player A keeps x secret, and truthfully tells N to player B.  Player B now tries to obtain information about x by asking player A questions as follows.  Each question consists of B specifying an arbitrary set S of positive integers (possibly one specified in a previous question), and asking A whether x belongs to S.  Player B may ask as many such questions as he wishes.  After each question, player A must immediately answer it with yes or no, but is allowed to lie as many times as she wishes; the only restriction is that, among any k+1 consecutive answers, at least one answer must be truthful.

After B has asked as many questions as he wants, he must specify a set X of at most n positive integers.  If x belongs to X, then B wins; otherwise, he loses.  Prove that:

  1. If n \geq 2^k, then B can guarantee a win.
  2. For all sufficiently large k, there exists an integer n \geq 1.99^k such that B cannot guarantee a win.
The comments to this post shall serve as the research thread for the project, in which participants are encouraged to post their thoughts and comments on the problem, even if (or especially if) they are only partially conclusive.  Participants are also encouraged to visit the discussion thread for this project, and also to visit and work on the wiki page to organise the progress made so far.
This project will follow the general polymath rules.  In particular:
  1. All are welcome. Everyone (regardless of mathematical level) is welcome to participate.  Even very simple or “obvious” comments, or comments that help clarify a previous observation, can be valuable.
  2. No spoilers! It is inevitable that solutions to this problem will become available on the internet very shortly.  If you are intending to participate in this project, I ask that you refrain from looking up these solutions, and that those of you have already seen a solution to the problem refrain from giving out spoilers, until at least one solution has already been obtained organically from the project.
  3. Not a race. This is not intended to be a race between individuals; the purpose of the polymath experiment is to solve problems collaboratively rather than individually, by proceeding via a multitude of small observations and steps shared between all participants.   If you find yourself tempted to work out the entire problem by yourself in isolation, I would request that you refrain from revealing any solutions you obtain in this manner until after the main project has reached at least one solution on its own.
  4. Update the wiki. Once the number of comments here becomes too large to easily digest at once, participants are encouraged to work on the wiki page to summarise the progress made so far, to help others get up to speed on the status of the project.
  5. Metacomments go in the discussion thread. Any non-research discussions regarding the project (e.g. organisational suggestions, or commentary on the current progress) should be made at the discussion thread.
  6. Be polite and constructive, and make your comments as easy to understand as possible. Bear in mind that the mathematical level and background of participants may vary widely.

Have fun!

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